The Subspace Emissary was designed in such a way that every character could make it through each stage. That combined with also being designed for multiplayer made Brawl's adventure mode feel like a copy of a copy of a generic level design.
Remember that ridiculous final jump in Samus' Board The Platforms on N64? Most of the characters couldn't make that jump and even Samus could just barely. The bonus stage was built around Samus' abilities. If an adventure mode is to work, it'd be best if every character had their own set of like five or six stages with regular free-for-all stages in between. That would be ideal though it's a **** ton of work and would likely only work single-player.
Otherwise, maybe if only certain characters could run through certain stages and play off their strengths. Kirby, King Dedede, and Pit have multiple jumps and are assigned a stage only they can get though, but Dedede is heavy, so he and Bowser are assigned a different stage that Kirby and Pit aren't. All the stages could be franchise mash-ups such as bits of Hyrule colliding with the Mushroom Kingdom. And multiplayer could work here since they would be designed around specific abilities and attributes rather than a character.
Also, every enemy should be an Assist Trophy so no more lame, forgettable, and generic foes. That also means there should be tons of them and some of them have to be a bit more sophisticated instead of being glorified Pokemon. They can be attacked and thrown. They can block, dodge, jump and grab ledges. On top of that, some should have more than one attack. Maybe you'll run into Ashley more than once. Work it into the story which still doesn't need (and preferably doesn't have) long cutscenes.